Question

Let V be the set of all functions that are twice differentiable in R and S={cosx,sinx,xcosx,xsinx}. a)Check that S is a linearly independent set over R.(Hint: Consider the equation a0cosx+a1sinx+a2xcosx+a3xsinx. Putx=0,π,π 2 ,π 4 ,etc.and solve for ai.) b) Let W=[S]and let T:V→V be the function defined by T(f(x))=d2 dx2(f(x))+2d dx(f(x)). Check that T is a linear transformation on V.

Answer 1

Answer on Question #67465 – Math – Linear Algebra

Question

Let $V$ be the set of all functions that are twice differentiable in $\mathbb{R}$ and

$$S = \{\cos x, \sin x, x \cos x, x \sin x\}.$$

a) Check that $S$ is a linearly independent set over $\mathbb{R}$. (Hint: Consider the equation

$$a_0 \cos x + a_1 \sin x + a_2 x \cos x + a_3 x \sin x.$$

Put $x = 0, \frac{\pi}{2}, \frac{\pi}{4}$ etc and solve for $a_i$.)

b) Let $V = [S]$ and let $T: V \to V$ be the function defined by $T\big(f(x)\big) = \frac{\partial^2}{\partial x^2} f(x) + 2\frac{\partial}{\partial x} f(x)$. Check that $T$ is a linear transformation on $V$.

Solution

a) $a_0 \cos x + a_1 \sin x + a_2 x \cos x + a_3 x \sin x = 0$.

For $x = 0$:

$$\begin{array}{l} a_0 \cos 0 + a_1 \sin 0 + a_2 \cdot 0 \cdot \cos 0 + a_3 \cdot 0 \cdot \sin 0 = 0 \rightarrow \\ a_0 + 0 a_1 + 0 a_2 + 0 a_3 = 0 \rightarrow \\ a_0 = 0. \end{array}$$

For $x = \pi$:

$$\begin{array}{l} a_1 \sin \pi + a_2 \cdot \pi \cdot \cos \pi + a_3 \cdot \pi \cdot \sin \pi = 0 \rightarrow \\ 0 a_1 - \pi a_2 + 0 a_3 = 0 \rightarrow \\ a_2 = 0. \end{array}$$

For $x = \frac{\pi}{2}$:

$$\begin{array}{l} a_1 \sin \frac{\pi}{2} + a_3 \cdot \frac{\pi}{2} \cdot \sin \frac{\pi}{2} = 0 \rightarrow \\ a_1 + \frac{\pi}{2} a_3 = 0 \rightarrow \\ a_1 = -\frac{\pi}{2} a_3. \end{array}$$

For $x = \frac{\pi}{4}$:

$$\begin{array}{l} \left(-\frac{\pi}{2} a_3\right) \sin \frac{\pi}{4} + a_3 \cdot \frac{\pi}{4} \cdot \sin \frac{\pi}{4} = 0 \rightarrow \\ \frac{2}{\sqrt{2}} \left(-\frac{\pi}{2} a_3\right) + \frac{\pi}{4} \cdot \frac{2}{\sqrt{2}} a_3 = 0 \rightarrow \\ -\frac{\pi}{2} a_3 + \frac{\pi}{4} a_3 = 0 \rightarrow \\ -\frac{\pi}{4} a_3 = 0 \rightarrow \\ a_3 = 0, \quad a_1 = -\frac{\pi}{2} a_3 = 0. \end{array}$$

Thus, over $\mathbb{R}$ the equation $a_0 \cos x + a_1 \sin x + a_2 x \cos x + a_3 x \sin x = 0$ has only the trivial solution $a_0 = a_1 = a_2 = a_3 = 0$, therefore $S$ is a linearly independent set over $\mathbb{R}$.

b) Linear transformation is such that

$$T(kf) = kT(f), \, k \in \mathbb{R}$$

and

$$T(f + g) = T(f) + T(g).$$

Since differentiation is a linear operation

$$(u + v)' = u' + v' \text{ and } (ku)' = ku',$$

transformation $T$ is also linear.

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